An iterative fast sweeping method for Eikonal equation in 2D anisotropic media on unstructured triangular meshes

Computation of traveltimes and ray paths is important for anisotropic tomography inversions. The Eikonal-equation-based method outperforms traditional ray methods by producing more accurate results. However, most existing Eikonal solvers are formulated on structured regular meshes, which are no longer accurate for models with the presence of irregular topography and subsurface interfaces. To solve Eikonal equation in vertically transversely isotropic (VTI) or tilted transversely isotropic (TTI) models with irregular geometry, we formulate a new iterative fast sweeping method on unstructured triangular meshes. The fixed-point iteration is implemented to capture the high-order nonlinear terms therein and a fast sweeping method on unstructured triangular meshes is implemented to solve the resulting elliptically anisotropic Eikonal equation at every iteration. We test the new algorithm for direct arrivals and reflected arrivals, and then use the calculated traveltimes to track the ray path in VTI/TTI media. Numerical tests demonstrate the validity and accuracy of the new method for models with rough topography and subsurface interface.

Microseismic Location Error Due to Eikonal Traveltime Calculation

The accuracy of computed traveltimes in a velocity model plays a crucial role in localization of microseismic events. The conventional approach usually utilizes robust fast sweeping or fast marching methods to solve the eikonal equation numerically with a finite-difference scheme. These methods introduce traveltime errors that strongly depend on the direction of wave propagation. Such error results in moveout changes of the computed traveltimes and introduces significant location bias. The issue can be addressed by using a finite-difference scheme to solve the factored eikonal equation. This equation yields significantly more accurate traveltimes and therefore reduces location error, though the traveltimes computed with the factored eikonal equation still contain small errors with systematic bias. Alternatively, the traveltimes can be computed using a physics-informed neural network solver, which yields more randomized traveltimes and resulting location errors.